I'm very far from being an expert at this sort of thing but I think I understand the bare mechanics of a claiming race. You enter your horse at a declared value and somebody, more or less anybody, has a right to buy at that price. Best way I understand, it is a way of keeping the owner from trying to slip in a ringer: he lowballs the value, he risks losing the horse.
I had a friend back in Kentucky who said we ought to use that device for value-based property taxation: the owner gets to declare its value and the sovereign can tax at that value, or buy at that price.
It struck me as an interesting idea though but while I can't say I've searched very hard, I've never before found anyone who took it very seriously (just now I tried it on a couple of my taxy friends who seem not to have given it any thought).
But maybe I have a contender. The subject for the moment is “sound dues.” We're at Kronberg Castle, facing out from the coast of Denmark towards Sweden just four miles away. The “dues” are the customs duties that the King of Sweden
extorted collected from passing ships,
starting in the 1420s and continuing until 1857 (the Americans
bullied him out of it).
You can see where this is going. So I'm told (I haven't seen a written source), the system operated on the principle of the claiming race. The captain would declare the value of his cargo; the king could tax or buy. I don't know any of the details: how long the system lasted; how many times the King exercised his right; how much money he made—nor, indeed, whether the story is a total fairy tail (the Danes do have a history with fairy tales). So, claiming-race taxation on the sound. There must be a whole literature out there that I haven't discovered yet but I must say I find the idea highly intriguing.